<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn> Assume the solution in the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=e^x (A \sin x+B \cos x).
\end{equation*}
</div>
<p class="continuation">Substituting it into the ODE, one has</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp;2 e^x (A \cos x-B \sin x)-2 e^x \left[(A-B) \sin x+ (A+B) \cos x \right]-3 e^x (A \sin x+B \cos x)=2 e^x \sin x\\
&amp; \to -5 e^x A \sin x-5 e^x B \cos x=2 e^x \sin x\\
&amp;\to -5A=2, \quad -5 B=0\\
&amp;\to A=-\frac{2}{5}, \quad B=0.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Thus, the particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
Y=-\frac{2}{5} e^x \sin x.
\end{equation*}
</div>
<span class="incontext"><a href="sec3_6.html#p-126" class="internal">in-context</a></span>
</body>
</html>
